climleltrp - Mathbox for Glauco Siliprandi

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Theorem climleltrp 43107

Description: The limit of complex number sequence 𝐹 is eventually approximated. (Contributed by Glauco Siliprandi, 26-Jun-2021.)

**Hypotheses**
Ref	Expression
climleltrp.k	⊢ Ⅎ𝑘𝜑
climleltrp.f	⊢ Ⅎ𝑘𝐹
climleltrp.z	⊢ 𝑍 = (ℤ_≥‘𝑀)
climleltrp.n	⊢ (𝜑 → 𝑁 ∈ 𝑍)
climleltrp.r	⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑁)) → (𝐹‘𝑘) ∈ ℝ)
climleltrp.a	⊢ (𝜑 → 𝐹 ⇝ 𝐴)
climleltrp.c	⊢ (𝜑 → 𝐶 ∈ ℝ)
climleltrp.l	⊢ (𝜑 → 𝐴 ≤ 𝐶)
climleltrp.x	⊢ (𝜑 → 𝑋 ∈ ℝ⁺)

**Assertion**
Ref	Expression
climleltrp	⊢ (𝜑 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℝ ∧ (𝐹‘𝑘) < (𝐶 + 𝑋)))

Distinct variable groups: 𝐴,𝑗,𝑘 𝑗,𝐹 𝑗,𝑁,𝑘 𝑗,𝑋,𝑘 𝑗,𝑍 𝜑,𝑗 Allowed substitution hints: 𝜑(𝑘) 𝐶(𝑗,𝑘) 𝐹(𝑘) 𝑀(𝑗,𝑘) 𝑍(𝑘)

**Proof of Theorem climleltrp**
Step	Hyp	Ref	Expression
1		climleltrp.n	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ 𝑍)
2		climleltrp.z	. . . . 5 ⊢ 𝑍 = (ℤ_≥‘𝑀)
3	1, 2	eleqtrdi 2849	. . . 4 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘𝑀))
4		uzss 12534	. . . 4 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → (ℤ_≥‘𝑁) ⊆ (ℤ_≥‘𝑀))
5	3, 4	syl 17	. . 3 ⊢ (𝜑 → (ℤ_≥‘𝑁) ⊆ (ℤ_≥‘𝑀))
6	5, 2	sseqtrrdi 3968	. 2 ⊢ (𝜑 → (ℤ_≥‘𝑁) ⊆ 𝑍)
7		climleltrp.k	. . . 4 ⊢ Ⅎ𝑘𝜑
8		climleltrp.f	. . . 4 ⊢ Ⅎ𝑘𝐹
9		uzssz 12532	. . . . 5 ⊢ (ℤ_≥‘𝑀) ⊆ ℤ
10	9, 3	sselid 3915	. . . 4 ⊢ (𝜑 → 𝑁 ∈ ℤ)
11		eqid 2738	. . . 4 ⊢ (ℤ_≥‘𝑁) = (ℤ_≥‘𝑁)
12		climleltrp.a	. . . 4 ⊢ (𝜑 → 𝐹 ⇝ 𝐴)
13		eqidd 2739	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑁)) → (𝐹‘𝑘) = (𝐹‘𝑘))
14		climleltrp.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ ℝ⁺)
15	7, 8, 10, 11, 12, 13, 14	clim2d 43104	. . 3 ⊢ (𝜑 → ∃𝑗 ∈ (ℤ_≥‘𝑁)∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑋))
16		nfv 1918	. . . . . 6 ⊢ Ⅎ𝑘 𝑗 ∈ (ℤ_≥‘𝑁)
17	7, 16	nfan 1903	. . . . 5 ⊢ Ⅎ𝑘(𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝑁))
18		simplll 771	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝑁)) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑋) → 𝜑)
19		uzss 12534	. . . . . . . . . . 11 ⊢ (𝑗 ∈ (ℤ_≥‘𝑁) → (ℤ_≥‘𝑗) ⊆ (ℤ_≥‘𝑁))
20	19	ad2antlr 723	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝑁)) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → (ℤ_≥‘𝑗) ⊆ (ℤ_≥‘𝑁))
21		simpr 484	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝑁)) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → 𝑘 ∈ (ℤ_≥‘𝑗))
22	20, 21	sseldd 3918	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝑁)) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → 𝑘 ∈ (ℤ_≥‘𝑁))
23	22	adantr 480	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝑁)) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑋) → 𝑘 ∈ (ℤ_≥‘𝑁))
24		simpr 484	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝑁)) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑋) → (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑋)
25		climleltrp.r	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑁)) → (𝐹‘𝑘) ∈ ℝ)
26	13, 25	eqeltrd 2839	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑁)) → (𝐹‘𝑘) ∈ ℝ)
27	26	adantr 480	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑁)) ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑋) → (𝐹‘𝑘) ∈ ℝ)
28		climcl 15136	. . . . . . . . . . . . . . 15 ⊢ (𝐹 ⇝ 𝐴 → 𝐴 ∈ ℂ)
29	12, 28	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐴 ∈ ℂ)
30	29	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑁)) → 𝐴 ∈ ℂ)
31	26	recnd 10934	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑁)) → (𝐹‘𝑘) ∈ ℂ)
32	30, 31	pncan3d 11265	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑁)) → (𝐴 + ((𝐹‘𝑘) − 𝐴)) = (𝐹‘𝑘))
33	32	eqcomd 2744	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑁)) → (𝐹‘𝑘) = (𝐴 + ((𝐹‘𝑘) − 𝐴)))
34	33	adantr 480	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑁)) ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑋) → (𝐹‘𝑘) = (𝐴 + ((𝐹‘𝑘) − 𝐴)))
35	34, 27	eqeltrrd 2840	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑁)) ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑋) → (𝐴 + ((𝐹‘𝑘) − 𝐴)) ∈ ℝ)
36		climleltrp.c	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐶 ∈ ℝ)
37	36	ad2antrr 722	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑁)) ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑋) → 𝐶 ∈ ℝ)
38	7, 8, 11, 10, 12, 25	climreclf 43095	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐴 ∈ ℝ)
39	38	ad2antrr 722	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑁)) ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑋) → 𝐴 ∈ ℝ)
40	27, 39	resubcld 11333	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑁)) ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑋) → ((𝐹‘𝑘) − 𝐴) ∈ ℝ)
41	37, 40	readdcld 10935	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑁)) ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑋) → (𝐶 + ((𝐹‘𝑘) − 𝐴)) ∈ ℝ)
42	14	rpred 12701	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑋 ∈ ℝ)
43	42	ad2antrr 722	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑁)) ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑋) → 𝑋 ∈ ℝ)
44	37, 43	readdcld 10935	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑁)) ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑋) → (𝐶 + 𝑋) ∈ ℝ)
45		climleltrp.l	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐴 ≤ 𝐶)
46	45	ad2antrr 722	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑁)) ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑋) → 𝐴 ≤ 𝐶)
47	39, 37, 40, 46	leadd1dd 11519	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑁)) ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑋) → (𝐴 + ((𝐹‘𝑘) − 𝐴)) ≤ (𝐶 + ((𝐹‘𝑘) − 𝐴)))
48	31	adantr 480	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑁)) ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑋) → (𝐹‘𝑘) ∈ ℂ)
49	30	adantr 480	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑁)) ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑋) → 𝐴 ∈ ℂ)
50	48, 49	subcld 11262	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑁)) ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑋) → ((𝐹‘𝑘) − 𝐴) ∈ ℂ)
51	50	abscld 15076	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑁)) ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑋) → (abs‘((𝐹‘𝑘) − 𝐴)) ∈ ℝ)
52	40	leabsd 15054	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑁)) ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑋) → ((𝐹‘𝑘) − 𝐴) ≤ (abs‘((𝐹‘𝑘) − 𝐴)))
53		simpr 484	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑁)) ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑋) → (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑋)
54	40, 51, 43, 52, 53	lelttrd 11063	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑁)) ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑋) → ((𝐹‘𝑘) − 𝐴) < 𝑋)
55	40, 43, 37, 54	ltadd2dd 11064	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑁)) ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑋) → (𝐶 + ((𝐹‘𝑘) − 𝐴)) < (𝐶 + 𝑋))
56	35, 41, 44, 47, 55	lelttrd 11063	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑁)) ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑋) → (𝐴 + ((𝐹‘𝑘) − 𝐴)) < (𝐶 + 𝑋))
57	34, 56	eqbrtrd 5092	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑁)) ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑋) → (𝐹‘𝑘) < (𝐶 + 𝑋))
58	27, 57	jca 511	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑁)) ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑋) → ((𝐹‘𝑘) ∈ ℝ ∧ (𝐹‘𝑘) < (𝐶 + 𝑋)))
59	18, 23, 24, 58	syl21anc 834	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝑁)) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑋) → ((𝐹‘𝑘) ∈ ℝ ∧ (𝐹‘𝑘) < (𝐶 + 𝑋)))
60	59	adantrl 712	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝑁)) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) ∧ ((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑋)) → ((𝐹‘𝑘) ∈ ℝ ∧ (𝐹‘𝑘) < (𝐶 + 𝑋)))
61	60	ex 412	. . . . 5 ⊢ (((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝑁)) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → (((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑋) → ((𝐹‘𝑘) ∈ ℝ ∧ (𝐹‘𝑘) < (𝐶 + 𝑋))))
62	17, 61	ralimdaa 3140	. . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝑁)) → (∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑋) → ∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℝ ∧ (𝐹‘𝑘) < (𝐶 + 𝑋))))
63	62	reximdva 3202	. . 3 ⊢ (𝜑 → (∃𝑗 ∈ (ℤ_≥‘𝑁)∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑋) → ∃𝑗 ∈ (ℤ_≥‘𝑁)∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℝ ∧ (𝐹‘𝑘) < (𝐶 + 𝑋))))
64	15, 63	mpd 15	. 2 ⊢ (𝜑 → ∃𝑗 ∈ (ℤ_≥‘𝑁)∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℝ ∧ (𝐹‘𝑘) < (𝐶 + 𝑋)))
65		ssrexv 3984	. 2 ⊢ ((ℤ_≥‘𝑁) ⊆ 𝑍 → (∃𝑗 ∈ (ℤ_≥‘𝑁)∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℝ ∧ (𝐹‘𝑘) < (𝐶 + 𝑋)) → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℝ ∧ (𝐹‘𝑘) < (𝐶 + 𝑋))))
66	6, 64, 65	sylc 65	1 ⊢ (𝜑 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℝ ∧ (𝐹‘𝑘) < (𝐶 + 𝑋)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1539 Ⅎwnf 1787 ∈ wcel 2108 Ⅎwnfc 2886 ∀wral 3063 ∃wrex 3064 ⊆ wss 3883 class class class wbr 5070 ‘cfv 6418 (class class class)co 7255 ℂcc 10800 ℝcr 10801 + caddc 10805 < clt 10940 ≤ cle 10941 − cmin 11135 ℤcz 12249 ℤ_≥cuz 12511 ℝ⁺crp 12659 abscabs 14873 ⇝ cli 15121

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-rep 5205 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879 ax-pre-sup 10880

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rmo 3071 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-iun 4923 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-om 7688 df-2nd 7805 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-er 8456 df-pm 8576 df-en 8692 df-dom 8693 df-sdom 8694 df-sup 9131 df-inf 9132 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-div 11563 df-nn 11904 df-2 11966 df-3 11967 df-n0 12164 df-z 12250 df-uz 12512 df-rp 12660 df-fl 13440 df-seq 13650 df-exp 13711 df-cj 14738 df-re 14739 df-im 14740 df-sqrt 14874 df-abs 14875 df-clim 15125 df-rlim 15126

This theorem is referenced by: smflimlem2 44194

W3C validator